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Two Particle system in inertial frames, and Newtonian Relativity.

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Consider an isolated system comprising two particles of masses m1 and m2. Whose position vectors, in an inertial frame, are x1 and x2 and whose velocity vectors are v1 and v2. The interaction of the particles may be described by an energy function.

E=1/2m1v1^2+1/2m2v2^2+U(x1,x2)

i)
Which aspect of Newtonian relativity requires U to depend only on the separation vector x1 - x2?

The question goes on to ask six more questions, two regarding this function and four regarding inertial frames and transformations between frames in general.

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The solution examines two particle systems in inertial frames and Newtonian Relativity.

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The answers are explained in the attached Word file which was created from the file you uploaded, with my remarks added in the red of "Track changes".

Question 1

Consider an isolated system comprising two particles of masses m1 and m2. Whose position

vectors, in an inertial frame, are x1 and x2 and whose velocity vectors are v1 and v2.

The interaction of the particles may be described by an energy function.

i)
Which aspect of Newtonian relativity requires U to depend only on the separation vector

x1 - x2?
Homogeneity of space: the laws of physics are the same everywhere

ii)
Which further aspect of Newtonian relativity requires U to depend only on the magnitude r

= | x1 - x2| of the separation vector?
Isotropy of space: all the directions are equivalent

iii)
Suppose that U = -k/r2, where k is a positive constant. ...

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